Elastic strain engineering of materials

ABSTRACT

Methods for training statistical models for modeling phononic energy and/or frequency dispersion, as well as phononic stability of materials as a function of an applied strain, as well as uses of these trained statistical models for elastic strain engineering of materials, are described.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit under 35 U.S.C. § 119(e) of U.S. provisional application Ser. No. 63/070,913, filed Aug. 27, 2020, the disclosure of which is incorporated by reference in its entirety.

GOVERNMENT FUNDING

This invention was made with Government support under Grant No. N00014-18-1-2497 awarded by the Office of Naval Research. The Government has certain rights in the invention.

FIELD

Disclosed embodiments are related to elastic strain engineering of materials.

BACKGROUND

Materials, including nano structured materials, have been developed that can withstand much higher tensile and/or shear elastic strains without mechanical relaxation or failure than their conventional counterparts. This range of possible strains open up a huge parameter space for engineering of material properties through the application of elastic strain. For example, strain can be used to tune various material properties analogous to the manipulation of the material's chemistry. For example, the electronic bandgap of a material may open or close with a particular strain, thus, altering the electrical, thermal, optical and/or magnetic characteristics of the material.

SUMMARY

Methods for training statistical models for modeling phononic energy and/or frequency dispersion, as well as phononic stability of materials as a function of an applied strain, as well as uses of these trained statistical models for elastic strain engineering of materials, are described.

In one aspect, a method is provided. In some embodiments, the method comprises, by at least one processor: obtaining a range of strain coordinates; obtaining training data, wherein the training data includes phononic energy and/or frequency dispersion training data for a material within the range of strain coordinates; generating a first statistical model using the training data and the range of strain coordinates, wherein the first statistical model is associated with the material and configured to output phononic energy and/or frequency dispersion values using at least one strain coordinate; and storing the first statistical model.

In another aspect, a method is provided. In some embodiments, the method, comprises, by at least one processor: obtaining a strain state of a component; using a first statistical model to estimate one or more phonon frequencies associated with the strain-state; and determining whether the strain state is a stable state using the estimated one or more phonon frequencies.

In yet another aspect, a method is provided. In some embodiments, the method comprises: providing a desired bandgap to a trained statistical bandgap model of a material to obtain one or more strain states corresponding to the desired bandgap; using a first statistical model to estimate one or more phonon frequencies associated with at least one of the one or more strain states; and determining whether at least one of the one or more strain states is a stable state using the estimated one or more phonon frequencies.

It should be appreciated that the foregoing concepts, and additional concepts discussed below, may be arranged in any suitable combination, as the present disclosure is not limited in this respect. Further, other advantages and novel features of the present disclosure will become apparent from the following detailed description of various non-limiting embodiments when considered in conjunction with the accompanying figures.

In cases where the present specification and a document incorporated by reference include conflicting and/or inconsistent disclosure, the present specification shall control. If two or more documents incorporated by reference include conflicting and/or inconsistent disclosure with respect to each other, then the document having the later effective date shall control.

BRIEF DESCRIPTION OF DRAWINGS

The accompanying drawings are not intended to be drawn to scale. In the drawings, each identical or nearly identical component that is illustrated in various figures may be represented by a like numeral. For purposes of clarity, not every component may be labeled in every drawing. In the drawings:

FIG. 1A is a schematic diagram showing the bandgap in an unstrained material;

FIG. 1B is a schematic diagram showing the bandgap of FIG. 1A after a strain is applied to the material;

FIG. 2A is a schematic diagram of strains that can be applied to a material, according to certain embodiments;

FIG. 2B is a schematic diagram of a strain being applied to deform a unit cell, according to certain embodiments;

FIG. 3A is one embodiment of a fitting process to determine a trained statistical model;

FIG. 3B is one embodiment of a fitting process to determine a trained statistical model;

FIG. 4 is another embodiment of a fitting process to determine a trained statistical model;

FIG. 5 is one embodiment of a fitting process to determine a trained statistical phononic energy and/or frequency dispersion model;

FIG. 6A is one embodiment of a neural network for bandgap fitting;

FIG. 6B is one embodiment of a neural network for band structure (i.e. energy dispersion) fitting;

FIG. 7 is a flow diagram of one embodiment for determining a stable strain state associated with a desired bandgap;

FIG. 8 is a flow diagram of one embodiment for determining the phononic stability of a strained component during finite element analysis;

FIG. 9 is a flow diagram of one embodiment of determining the phononic stability and, optionally, the electron energy dispersion and/or one or more material properties of a material using a trained statistical model;

FIG. 10 presents a schematic illustration of a phononic stability envelope within a cross-section of a strain-space, according to certain embodiments;

FIG. 11 is a schematic embodiment of a system for training and/or implementing the models disclosed herein ;

FIG. 12 presents exemplary phonon density of state functions, according to some embodiments;

FIG. 13A-13B present cross-sections of a phonon stability region within a strain-space for an exemplary material, according to some embodiments;

FIG. 14 presents exemplary cross-sections of a phononic stability envelope in a six dimensional strain space, according to certain embodiments;

FIG. 15 presents a schematic illustration of an application of multiaxial strain to a material using an actuator, according to certain embodiments;

FIG. 16A presents a multiaxial stress state of a nanoneedle, in some embodiments;

FIG. 16B presents a multiaxial stress state of a nanoneedle, in some embodiments;

FIGS. 17A-17B present minimum electronic bandgap energy of an exemplary material as a function of strain, according to some embodiments; and

FIG. 18 presents a minimum electronic bandgap energy of exemplary nanoneedles and a volume of a direct bandgap region as a function of strain, according to some embodiments.

DETAILED DESCRIPTION

The Inventors have recognized that predicting and/or testing the material properties of a strain engineered material with strains having three or more degrees of freedom is extrememly difficult due to the much larger at least three-dimensional hyperspace that may be investigated. Accordingly, it is impractical to experimentally explore this entire range of possible strains that may be applied to a material to alter the properties of that material. Also, due to the computational complexity involved in determining the phononic stability of a material at a particular strain, it is also impractical to calculate all of the possible combinations of strain and phonon dispersion of a material. The Inventors have recognized that this task is further complicated due to possible phase changes caused by the applied high levels of strain resulting in thermodynamically unstable states which may cause unwanted permenant material and property changes, and in some cases failure of a device.

In view of the above, the Inventors have recognized the benefits associated with maintaining a material within a strain region where it is thermodynamically stable without undergoing an irreversable phase change and/or permenant deformation or fracture. Thus, the Inventors have developed a trained statistical model for determining the relationship between phononic stability of a material versus strain having three or more, and in some instances at least six degrees of freedom. This may include the application of three-dimensional and/or six-dimensional strain tensors to a material. Specifically, conventional methods of calculating phonon frequency and/or energy dispersion of a material may be conducted to obtain a desired number of training data points. Alternatively, in some embodiments, the training data may be obtained from data available from prior experiments and/or calculations. This training data may be input into a statistical model along with a desired range of strain coordinates to generate a trained statistical model. Again these strain coordinates may include strains with at least three or more degrees of freedom, and in some embodiments six degrees of freedom. In some embodiments, the model may be used along with a trained statistical bandgap model, e.g., in order to allow identification of stable strain states with a desired bandgap.

The above detailed concept of generating a statistical model for predicting the phononic stability and/or phonon dispersion of a material for strains is very general in nature. For example, the process may be used for any number of different types of crystals and materials as well as across any number of different desired ranges of strains with any number of desired degrees of freedom for the applied strains including, for example, three-dimensional strains, six-dimensional strains, and/or any other appropriate strain with any appropriate degrees of freedom. The statistical model may be trained, e.g., using a range of strain coordinates and associated training data. In some embodiments, the statistical model is a machine learning model (e.g., a neural net model) as described in greater detail below.

In view of the above, in some embodiments a trained statistical model (e.g., associated with a material) is generated. Once a trained statistical model has been generated, the trained statistical model may be stored for subsequent use. For example, the trained statistical model may be stored on at least one non-transitory computer readable storage medium. In some embodiments, storing the trained statistical model comprises storing weights of a machine learning model (e.g., a neural network model). The trained statistical model may be configured to output phononic energy and/or frequency dispersion values. For example, the trained statistical model may be configured to output phononic energy and/or frequency dispersion values using at least one strain coordinate (a strain coordinate may also be referred to herein as a strain state). In some embodiments, the trained statistical model may be configured to directly output a phononic stability state. For example, the trained statistical model may be configured to output a phononic stability state using at least one strain coordinate.

The stored model may then be used for a number of different applications related to determining the properties of a material under a strain as detailed further below. These uses may include, but are not limited to, determining phononic stability of strain states of a material corresponding to a desired bandgap of the material, visualization of phonon band structures of a material with a particular strain state, identification of the lowest energy density strain state that is phononically stable and provides a desired bandgap, use with finite element analysis modules to determine the stability of a strained material, and/or any other appropriate use as described herein.

It should be understood that the strains disclosed herein for either the training and/or use of a trained statistical model may correspond to any desired range of strains, and may be assumed to refer to elastic strains unless otherwise indicated, where an elastic strain refers to a recoverable strain rather than a permeant irrecoverable plastic strain. For example, in some embodiments, a range of strain coordinates associated with a trained statistical model may correspond to the physical elastic strain limits of a material being modeled. Alternatively, the range of strain coordinates may be a subset of this physically possible range as selected and determined by a user, a preset range of strain coordinates, and/or any other appropriate method for determining a range of strain coordinates for modeling purposes. That said, in one embodiment, each of the components of a strain tensor may be independently selected to be an elastic strain that is greater than −20%, −15%, −10%, −5%, 0%, and/or any other appropriate strain. Correspondingly, each component of the strain tensor may be independently selected to be less than 20%, 15%, 10%, 5%, and/or any other appropriate strain. Combinations of the forgoing are contemplated including, for example, each of the individual strain components may be between or equal to −20% and 20%, −10% and 10%, and/or any other appropriate range of strains. In addition to the individual applied strain components, an overall magnitude of an applied strain may be between or equal to 1% and 20%, 1% and 10%, 5% and 20%, 5% and 10%, and/or any other appropriate range of strains as the disclosure is not so limited.

For the purposes of the various embodiments and examples described herein, a strain, strain coordinate, strain tensor, or other similar term may refer to a strain that has been applied to a material with three or more degrees of freedom including, for example three degrees of freedom (i.e. three-dimensional strain space), six degrees of freedom (i.e. six-dimensional strain space), and/or any other appropriate strain, strain tensor, or strain coordinate applied with any desired number of degrees of freedom. Additionally, these strains may also refer to elastic strains. For example, a shearless elastic strain may be applied using a three-dimensional strain tensor with three degrees of freedom while in another embodiment due to the strain tensor being a symmetric matrix non-redundant strains may be defined using six degrees of freedom resulting in what may be described as a six-dimensional strain tensor. Accordingly, it should be understood that strains may be described relative to the disclosed embodiments using any appropriate strain with any appropriate number of degrees of freedom ranging from three degrees of freedom strain (three-dimensional strain) to six degrees of freedom strain (six-dimensional strain) as the disclosure is not limited in this fashion. A more detailed description of strain is provided below.

Using the models described herein, particular strain states of silicon associated with various material property transitions were identified as discussed further in the examples. For example, in one embodiment, the least strain energy density to transition silicon from an indirect to a direct bandgap material may occur for a triaxial tensile strain applied to a silicon material defined by the strain tensor:

$\quad\begin{pmatrix} ɛ_{11} & 0 & 0 \\ 0 & ɛ_{22} & 0 \\ 0 & 0 & ɛ_{33} \end{pmatrix}$

where ε₁₁, ε₂₂, and ε₃₃ are between or equal to 8.9% and 9.5%, and in some instances these strain components may preferably be approximately 9.2%. Further in some embodiments the strain components may be approximately equal to each other.

In another embodiment, the least energy required to realize a semiconductor/insulator to metal transition in silicon is by applying an elastic strain tensor of

$\quad\begin{pmatrix} ɛ_{11} & ɛ_{12} & ɛ_{13} \\ ɛ_{21} & ɛ_{22} & ɛ_{23} \\ ɛ_{31} & ɛ_{32} & ɛ_{33} \end{pmatrix}$

where ε₁₁ is between or equal to 0.25% and 0.75%, ε₁₂ is between or equal to 1.45% and 1.95%, ε₁₃ is between or equal to 0.5% and 1.0%, ε₂₁ is between or equal to 1.45% and 1.95%, ε₂₂ is between or equal to −1.0% and −1.5%, ε₂₃ is between or equal to −1.65% and −2.15%, ε₃₁ is between or equal to 0.5% and 1.0%, ε₃₂ is between or equal to −1.65% and −2.15%, and ε₃₃ is between or equal to −0.8% and −1.35%. In some embodiments, these strain components may more preferably correspond to ε₁₁ being approximately 0.55%, ε₁₂ being approximately 1.69%, ε₁₃ being approximately 0.74%, ε₂₁ being approximately 1.69%, ε₂₂ being approximately −1.26%, ε₂₃ being approximately −1.92%, ε₃₁ being approximately 0.74%, ε₃₂ being approximately −1.92%, and ε₃₃ is approximately −1.04%.

Due to the desirable properties afforded by the material property transitions noted above for the strain engineered silicon, it should be understood that strained silicon with the above noted strain states may be included in an electrical circuit. For example, an electrical component of an electrical circuit may be formed from silicon with a strain tensor as described above. This electrical component may be electrically connected to any other appropriate portion of the electrical circuit to form a functional part of the circuit including, but not limited to, electrical contact pads, junctions, electrical traces, and/or any other appropriate electrical component of a circuit as the disclosure is not so limited.

In some embodiments, the components made from an elastically strained material may have a characteristic length scale to enhance elastic strain limit of the material. For example, certain nanoscale materials may exhibit enhanced elastic strain limits as compared to larger bulk materials. Accordingly, a component made from a material that has been elastically strained to modify one or more material properties as described herein may have a characteristic length scale, such as a thickness, or other appropriate dimension, that is less than about 1 μm, 500 nm, 100 nm, and/or any other appropriate length scale. Appropriate types of materials that may be elastically strained to modify their material properties may include, but are not limited to, silicon, diamond, gallium arsenide (GaAS), germanium (Ge), gallium nitride (GaN), two-dimensional/atomically thin materials, and/or any other appropriate material. Of course, it should be understood that components with characteristic length scales that are both larger and smaller than those noted above, as well as components that are made using different materials are also contemplated as the disclosure is not limited in this fashion.

Again, it is difficult to determine experimentally determine and correlate both the stability and bandgap of a material for all of the points within the possible strain states of a material. Accordingly, as noted above, the Inventors have recognized the benefits associated with using a trained statistical phononic energy and/or frequency dispersion model alongside a trained statistical model for determining the relationship between the electron bandgap and/or the electron energy dispersion of a material versus strain. Thus, in some embodiments, a bandgap corresponding to a stable strain state of the component using a second statistical model is obtained. One advantage of this approach may be identifying a corresponding stability state when calculating the bandgap and/or energy dispersion of a material, e.g., to determine whether the calculation corresponds to a physically stable situation that can be actually be physically realized. As with determination of phononic properties, trained statistical models may be used to calculate bandgap and/or electron energy dispersion of a material. Conventional methods of calculating the bandgap and/or electron energy dispersion of a material may be conducted to obtain a desired number of training data points. Alternatively, in some embodiments, the training data may be obtained from data available from prior experiments and/or calculations. This training data may be input into a statistical model (e.g., a second statistical model) along with a desired range of strain coordinates to generate a trained statistical model. Again these strain coordinates may include strains with at least three or more degrees of freedom, and in some embodiments six degrees of freedom. Depending on the particular embodiment, the trained statistical model may be either a bandgap model or an electron energy dispersion model. In instances in which the trained statistical model is an electron energy dispersion model, the training data and the desired range of strain coordinates as well as a corresponding range of reciprocal space coordinates may be provided to the statistical model which may output the desired trained statistical model.

As with statistical models of phonons, the above detailed concept of developing a statistical model for predicting the bandgap and/or energy dispersion of a material for strains is very general in nature. For example, the process may be used for any number of different types of crystals and materials as well as across any number of different desired ranges of strains with any number of desired degrees of freedom for the applied strains including, for example, three-dimensional strains, six-dimensional strains, and/or any other appropriate strain with any appropriate degrees of freedom. The stored statistical model may then be used for a number of different applications related to determining the properties of a material under a strain as detailed further below. These uses may include, but are not limited to, determining possible strain states of a material for a desired bandgap, visualization of band structures of a material with a particular strain state, identification of direct to indirect bandgap transitions, identification of semiconductor to metal bandgap transition, the lowest energy density strain state to provide a desired bandgap, use with finite element analysis modules to determine the bandgap of a strained material, and/or any other appropriate use as described herein.

For the purposes of this disclosure, the electron energy dispersion, which may also be referred to as the electron band structure, of a material may describe the range of energies that an electron within a material may have (i.e. electron energy bands, allowed electron energy states, electron bands, or other similar terms) and ranges of energy that the electrons do not occupy (i.e. bandgaps). These electron energy bands may be analyzed to identify various properties of a material as detailed further herein.

Additional details regarding the development and implementation of a trained statistical model for determining the electron energy dispersion of a material as a function of strain is further detailed in International Patent Application No. PCT/RU2018/000679 filed on Oct. 12, 2018, and published as WO 2020/076181, the disclosure of which is incorporated herein by reference in its entirety for all purposes.

As noted above, in some embodiments, it may be desirable to calculate an energy and/or frequency dispersion relation (also referred to herein as an energy and/or frequency dispersion) for phonons and/or electrons. A dispersion relation as described herein relates a wave-vector of a wave-propagating species (such as a phonon or an electron) to an energy and/or a frequency of the wave-propagating species. Thus, determining an energy and/or frequency dispersion relation may be equivalent to determining a mapping between the wave-vector of the wave-propagating species and the energy of the wave-propagating species (or alternatively, determining a mapping between the wave-vector of the wave-propagating species and the frequency of the wave-propagating species). In some embodiments, the energy and/or frequency dispersion of electrons and/or phonons is a property of a medium. For instance, an electronic energy and/or frequency dispersion may depend, at least in part, on a periodicity of an atomic arrangement distribution of the medium. As another example, a phononic energy and/or frequency dispersion may depend, at least in part, on a bond-strength of interatomic bonds (e.g., covalent bonds, ionic bonds, metallic bonds) in the medium.

Turning to the figures, specific non-limiting embodiments are described in further detail. It should be understood that the various systems, components, features, and methods described relative to these embodiments may be used either individually and/or in any desired combination as the disclosure is not limited to only the specific embodiments described herein.

FIGS. 1A and 1B illustrate several types of changes that may occur to a band structure of a material when an elastic strain is applied to the material. Specifically, FIG. 1A depicts a graph including a conduction band 100, a valence band 102, and a bandgap 104. In the illustrated embodiment, the material has a direct bandgap between the conduction band and the valence band. FIG. 1B illustrates a possible band structure for the same material after an elastic strain has been applied to the material. In the illustrated embodiment, the bandgap between the conduction and valence bands has decreased. Additionally, the relative positioning of the conduction band minimum and the valence band maximum has shifted so that the material is now an indirect bandgap material. Accordingly, the applied elastic strain has been used to change the properties of the material. However, as previously discussed, it is difficult to determine the specifics about how these properties will change with elastic strain for more than a few isolated strain coordinates due to the complex computations involved with solving such a problem when the strains are applied in at least three-dimensional strain space including when strain tensors and/or coordinates having at least three, six, or other appropriate number of degrees of freedom are applied.

FIG. 2A illustrates a cubic unit cell 200. The unit cell has diamond cubic crystallographic directions [100], [010], and [001] corresponding to the x, y, and z axes, respectively. A strain tensor may correspond to the various normal and shear strain components that may be applied to the unit cell. Typically, an elastic strain tensor may be indicated by:

${{ɛ =}\quad}\begin{pmatrix} ɛ_{11} & ɛ_{12} & ɛ_{13} \\ ɛ_{21} & ɛ_{22} & ɛ_{23} \\ ɛ_{31} & ɛ_{32} & ɛ_{33} \end{pmatrix}$

where ε₁₁, ε₂₂, ε₃₃ correspond to the strain components that are applied normal to the unit cell in the [100], [010], and [001] crystallographic directions respectfully. The remaining strain components, as indicated in the figure, are applied in directions parallel to the various surfaces of the unit cell and correspond to shear strains applied parallel to the surfaces of the unit cell. For the purposes of the current disclosure, the various strain tensors described herein may correspond to the above noted nomenclature. FIG. 2B illustrates a strain tensor being applied to elastically deform a unit cell of a crystal structure from an undeformed to a deformed configuration. For example, a strain tensor of:

${{ɛ =}\quad}\begin{pmatrix} {10\%} & 0 & 0 \\ 0 & {10\%} & 0 \\ 0 & 0 & {10\%} \end{pmatrix}$

refers to a strain coordinate corresponding to a 10% triaxial tension of the material, and:

${{ɛ =}\quad}\begin{pmatrix} 0 & {10\%} & 0 \\ {10\%} & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}$

is a strain coordinate corresponding to a 10% pure shear strain being applied to the material.

FIGS. 3A-5 depict embodiments related to methods for training statistical models of the phononic energy and/or frequency dispersion of a material relative to a desired range of strains.

In the embodiment illustrated in FIGS. 3A-3B, a range of strain coordinates may be obtained as described herein. The range of strain coordinates and training data may be used to generate a first statistical model. The range of strain coordinates may correspond to the physically possible range of elastic strains that may be applied to a material. Alternatively, a user may input a desired range of elastic strains over which a model of phononic stability of the material is desired. For example, a user may provide a desired elastic strain magnitude, absolute strain tensor component limits, and/or any other desirable combination thereof as the disclosure is not limited in this fashion. In such an embodiment, the range of strain coordinates may be provided by the user using any appropriate input device as described further below. However, embodiments in which the range of strain coordinates are predetermined for use in training a statistical model are also contemplated.

In addition to the range of strain coordinates, training data may also be used to generate the first statistical model. Specifically, phononic energy and/or frequency dispersion training data for a material within the range of strain coordinates may be obtained and used to generate the first statistical model (e.g., using a machine learning module), as shown in FIG. 3A. In some embodiments, training data may comprise phononic stability training data, as shown in FIG. 3B. In some embodiments, the training data comprise a plurality of training data points. Each training data point may be associated with a stress state (i.e., a stress coordinate) of the material. Each training data point may be associated with an energy and/or frequency value (e.g., in the case of phononic energy and/or frequency dispersion training data). In some embodiments, each training data point may be associated with a phononic stability state (e.g., as an alternative to, or in addition to, an energy and/or frequency value). Each training data point may be associated with a density of states (DOS) value for the energy and/or frequency value in some embodiments. Alternatively, in some embodiments, the training data, and eventual output from a statistical model, may include phonon frequencies instead. Each training data point may be associated with a point and/or a range of reciprocal space.

The training data may be obtained in any appropriate fashion using prior experiments and/or calculations. However, in some embodiments, obtaining the training data may correspond to performing ab-initio calculations to obtain the phononic energy and/or frequency dispersion training data or phononic stability training data prior to providing the training data to the first machine learning module. Due to the computationally expensive process of calculating this training data, the training data may be limited to a predetermined number of data points. For example, the training data may be limited to between or equal to 500 data points and 1000 data points, 500 data points and 2000 data points, and/or any other appropriate number of data points. Regardless of the specific number, these training data points may be randomly selected throughout the range of strain coordinates, evenly distributed throughout the range of strain coordinates, and/or any other appropriate disposition as the disclosure is not limited in this fashion. Regardless of how the training data points are distributed, in some embodiments, the training data may be calculated using first-principles computations, including standard density functional theory using a finite displacement method (FDM) or density functional perturbation theory (DFPT), and/or with any other appropriate methodology as the disclosure is not limited to how the bandgap and energy dispersion training data versus a desired strain space is obtained.

As also shown in FIG. 3A, the phononic energy and/or frequency dispersion training data as well as the range of strain coordinates may be used to generate the first statistical model (e.g., using a first machine learning module). Once these inputs have been received, a trained statistical model may be generated using the training data and the range of strain coordinates. The resulting output may be a trained statistical model of the phononic energy and/or frequency of a material as a function of the strain applied to the material, or may be an estimate of phononic stability as a function of the strain applied to the material. The first model may also be stored in an appropriate non-transitory computer readable medium for subsequent use as detailed further below.

Alternatively, as shown in FIG. 3B, the phononic stability training data as well as the range of strain coordinates may be used to generate the first statistical model (e.g., using a first machine learning module). Once these inputs have been received, a trained statistical model may be generated using the training data and the range of strain coordinates. The resulting output may be a trained statistical model of the phononic stability of a material as a function of the strain applied to the material. A trained statistical model that directly predicts phononic stability may be advantageous in some embodiments, when other information about phonon dispersion and/or density of states is not needed and could slow the process of computation.

In some embodiments, when generating an initial trained statistical model as detailed above, a machine learning module may implement a fitting approximation that is relatively less computationally expensive to permit the trained statistical model to be generated in a reasonable time period. For instance, in some embodiments the first trained statistical may be a phonon stability estimation and the second trained statistical model may estimate phononic dispersion. However, this may result in a trained statistical model that is less accurate than may be desired for certain applications. In such an instance, it may be desirable to generate a second trained statistical model that further refines the already trained statistical model using a second type of fitting approximation to generate a more accurate second trained statistical model. This process is sometimes referred to as delta machine learning and/or data fusion where a previously trained model is used as an input to a more accurate and computationally complex model to provide a more accurate model at a lower computational cost. Such an embodiment is depicted in FIG. 4, where a first trained statistical model corresponding to a first phonon energy and/or frequency dispersion model is generated and stored as detailed above relative to FIG. 4. This first phonon energy and/or frequency dispersion model may then be input along with the previously obtained training data and range of strain coordinates into a second machine learning module. The second machine learning module may then fit the training data using the first phonon energy and/or frequency dispersion model as a starting point and a more accurate, and correspondingly more computationally expensive or complex, fitting approximation. After the fitting process, the second machine learning module may output a second trained statistical model corresponding to the illustrated second phonon energy and/or frequency dispersion model. As above, the second phonon energy and/or frequency dispersion model may be stored for subsequent usage.

FIG. 5 depicts an embodiment similar to those described relative to FIGS. 3-4. However, in this particular embodiment, the generation of a phononic energy and/or frequency dispersion model that may be used to model the phononic energy bands and/or predict various properties of a material is depicted. Similar to the above embodiments, phononic energy and/or frequency dispersion training data may be obtained within a desired range of strain space coordinates. Correspondingly, a range of reciprocal space coordinates associated with the training data and/or range of strain space coordinates may also be obtained. A range of reciprocal space coordinates as described herein may in some embodiments correspond to a Fourier transform of the corresponding strain coordinates. The training data as well as the range of reciprocal space coordinates and the range of strain space coordinates may be input into a first machine learning module which may fit the phonon energy dispersion of the material versus strain. The resulting output is the generation of a trained strained statistical model that may be a phononic energy and/or frequency dispersion model which may be stored for subsequent usage on an appropriate non-transitory computer readable medium. While the usage of only a single machine learning module is depicted in the figure, it should be understood that embodiments in which an iterative machine learning process is implemented are also contemplated. For example, in some embodiments, the generated phononic energy and/or frequency dispersion model may be provided as an input to a second machine learning module using a more complex fitting approximation to generate a second more refined phononic energy and/or frequency dispersion model, as represented in FIG. 4B.

It should be understood that the trained statistical models disclosed herein may be generated using any appropriate statistical model. For example, the machine learning modules depicted in FIGS. 3-5, may correspond to any appropriate fitting method capable of generating the desired trained statistical models including, but are not limited to, artificial neural networks (ANN), gradient boosting regression (GBR), random forest regression (RFR), other kernel-based fitting methods, Lagrange polynomial fitting, and/or any other appropriate type of fitting method.

It should also be understood that the above noted fitting methods may be combined with any appropriate type of fitting approximation to provide a desired combination of model accuracy versus computational expense. For example, appropriate approximation methods that may be used include, but are not limited to, GW theory, HSE06, generalized gradient approximation, local density approximation, meta-GGA, combinations of the forgoing, and/or any other appropriate type of approximation as the disclosure is not limited in this fashion. Additionally, as noted previously, in instances where an iterative training process is used (i.e., data fusion), as shown in FIG. 4, the first statistical training model may use a first fitting approximation with a lower computational cost and the second statistical training model may use a second fitting approximation that is more accurate, but more computationally costly, then the first fitting approximation. In one such embodiment, an artificial neural network may use a first fitting approximation such as PBE to generate the first trained statistical model and a fitting approximation such as GW may be used with the artificial neural network to generate the second trained statistical model while using the first trained statistical model as an input. Of course, it should be understood that any appropriate combination of fitting approximations with the disclosed statistical models may be used as the disclosure is not limited to only the specifically disclosed combination.

FIGS. 6A-6B illustrate the structure of two possible embodiments of neural networks that may be used to generate the trained statistical models disclosed herein. FIG. 6A shows a deep neural network, including four hidden layers. The layers have a (64-128-256-256) structure. FIG. 6B illustrates a similar deep neural network including four hidden layers with a (512-256-256-256) structure. Without wishing to be bound by theory, the first neural network shown in FIG. 6A may be more suited to developing models for more limited ranges of strain, such as determining a phononic energy and/or frequency dispersion model for non-shear strains in using strains with three degrees of freedom, i.e. a three-dimensional strain tensor. In contrast, the more complex artificial neural network shown in FIG. 6B may be appropriate for use with more complex problems such as the modeling of the phononic energy and/or frequency dispersion of a material including shear strains, as well as the complicated task of developing a phononic energy and/or frequency dispersion model for predicting the phononic stability of a material strained using a strain tensor with three or more degrees of freedom. Depending on the particular embodiment, the deep neural networks depicted in the figures may be a feed-forward-structured artificial neural network including a leaky rectified linear unit as the activation function. The depicted artificial networks may also incorporate the Adam stochastic optimization method, the orthogonal weight initialization, and dropout technique to prevent overfitting. Of course, it should be understood that any appropriate structure for an artificial neural network may be used including artificial neural networks with any appropriate number of hidden layers and/or individual neurons per layer that are both greater than and less than those noted above as the current disclosure is not limited to only using the depicted artificial neural networks.

FIGS. 7-9 depict several possible uses of a trained statistical model that has been generated for predicting phononic energy and/or frequency dispersion of a material as a function of strains applied to the material. Of course, while several possible uses of a trained statistical model are provided, it should be understood that the currently disclosed trained statistical models are not limited to only these uses and that they may be generally applicable to any use where it is desirable to predict the phononic stability, the phononic band structure, the phonon density of states, and/or any other appropriate property related thereto as a function of a strain applied to a material. Many optional steps related to calculation of electron bandgaps and related properties are included in FIGS. 7-9, but are marked as optional using dashed lines and in-text labeling.

As noted previously, in some instances, the materials for which the currently disclosed trained statistical models may be used may be nanomaterials with characteristic length scales that are less than about 1μm. Although nanomaterials may be able to support relatively larger elastic strains without inducing plasticity and/or fracture for comparably longer times as compared to macroscale materials, it is still possible at higher strains and/or temperatures, that the applied elastic strains may relax away due to a variety of relaxation mechanisms. The physical limitations also apply to microscale materials, though the relative elastic strains that may be supported may be correspondingly lower as compared to nanoscale materials. Accordingly, in some embodiments, it may be desirable to identify elastic strain coordinates within a range of possible strains that provide a desired bandgap with a minimum corresponding amount of elastic strain energy density. This may help to reduce the possibility of fracture and/or relaxation of the strain of an elastic strain engineered material over time.

FIG. 7 depicts one embodiment of a method 300 of using a trained statistical model to identify one or more strains of a material with a bandgap that is less than or equal to a desired bandgap. Specifically, at 302 a desired bandgap is obtained. This bandgap may be obtained either by input from a user through an appropriate input device, the bandgap may be provided as a design parameter for a particular application, and/or any other appropriate source as the disclosure is not limited in this fashion. At 304, the desired bandgap may then be provided as an input to a trained statistical bandgap model of a material which, as noted previously, may be a function of the bandgap and strain. The trained statistical bandgap model may output at least one or more strain coordinates that, when applied, will result in the material exhibiting the desired bandgap at 306.

At step 307, a first trained statistical model (e.g., a phononic energy dispersion model) may be used to estimate one or more phonon frequencies associated with at least one of the one or more strain states. In some embodiments, it may be desirable to identify whether a strain state is stable (i.e., to determine a stability state of the strain state). The one or more phonon frequencies associated with the strain state may be used to determine whether the strain state is stable or unstable. Determining whether the strain state is stable or unstable (i.e., determining the stability state of the strain state) may comprise determining whether any of the estimated one or more phonon frequencies has a non-real or negative value and determining that the strain state is a stable state if none of the estimated one or more phonon frequencies has a non-real or negative value. In other words, if all of the estimated one or more phonon frequencies are positive real numbers, the strain state may be determined to be a stable strain state of the material. In some embodiments, a method comprises determining at least one phonon density of states (DOS) value associated with one of the estimated one or more phonon frequencies. Determining that the DOS is nonzero for non-real or negative frequencies and/or energies may indicate that a strain state is unstable, or stated in another way, if the DOS is zero for non-real or negative frequencies the strain state may be determined to be stable. In the above embodiments, the trained statistical model may output the DOS and/or the phonon frequencies as the output for use in the above comparisons.

In some embodiments, it may be desirable to identify a stable strain coordinate with the lowest corresponding strain energy density at 308 that is both stable and provides the desired bandgap energy. In some instances, this may be done by evaluating the set of strain coordinates using any appropriate method including following the steepest descent direction of the calculated strain energy densities associated with the output strain coordinates. Alternatively, in some embodiments, the trained statistical bandgap model may be used to provide gradient and/or curvature information related to a surface formed by the strain coordinates with the same bandgap. In some embodiments, the stable strain state with the lowest strain energy is identified by identifying a strain coordinate with the lowest corresponding strain energy density and subsequently confirming its stability. However, in some embodiments, the first trained statistical model (e.g., the phonon energy and/or frequency dispersion model) may be used to provide stability information about the strain state (e.g., a stability envelope as described in greater detail below), which may be incorporated directly into identifying the stable strain state with the lowest strain energy. For example, the stability envelope may be treated as an additional boundary constraint in a steepest descent method, or in any other suitable optimization method such that the method identifies a strain state that minimizes the strain energy density while maintaining the strain state within the stability envelope of thermodynamically stable strain states using the relationships detailed above.

Regardless of how the identification of a stable strain state with the lowest strain energy density and a desired bandgap is accomplished, after identifying the strain state, the set of strain states and/or the identified stable strain state with a lowest strain energy density for the desired bandgap may be stored in a non-transitory computer readable medium at 310 for subsequent recall and/or use. Additionally, in some embodiments, at 312 the set of strain coordinates and/or the identified strain coordinate may be output to a user. For example, the set of strain coordinates may be plotted as an isosurface in three-dimensional strain space where each point on the surface has the desired bandgap for visualization purposes by the user. Such a graph may help to visualize the design envelope for strain that the user may work within when designing a component with a desired material property. Alternatively, and/or a combination with this type of output graph, the strain coordinates corresponding to the lowest strain energy density for the desired bandgap may be output to the user as well, either numerically and/or in graphical form.

Due to the complexity and cost associated with the design and manufacture of microelectronic components, it may be desirable to evaluate the stability and/or other physical parameters of a material in view of the stresses and strains applied to those materials both when initially manufactured and/or during operation. Accordingly, in some embodiments, it may be desirable to use the disclosed phononic energy and/or frequency dispersion models described herein in combination with a Finite Element Analysis module for evaluating the resulting properties of material due to strains applied to the material prior to and/or during operation. One such method 400 is shown in FIG. 8 in relation to the use of a phononic energy and/or frequency dispersion model with a Finite Element Analysis module. However, it should be understood that any of the trained statistical models disclosed herein, including an electron energy dispersion model and/or an electron bandgap model may also be used in combination with a Finite Element Analysis module as the disclosure is not so limited.

In the depicted method 400, a model including the geometry and material properties of a component may be obtained at 402. Additionally, in instances where it is desirable to evaluate the material properties of the component during operation, one or more operational parameters of the component, and/or an associated system, may be obtained at 404. Appropriate types of operational parameters may include, but are not limited to, heat generation, loading, and/or the appropriate operational parameters. The model may be meshed at 406 using any appropriate mesh strategy to form a plurality of mesh elements. The meshed model including a plurality of mesh elements may be input along with the provided material properties and operational parameters to an associated Finite Element Analysis module at 408. It should be understood than any appropriate method of conducting a Finite Element Analysis may be used as the disclosure is not limited in this fashion. In either case, the strain states for the individual mesh elements may be obtained at 410, using the Finite Element Analysis module.

Once the strain states of the individual mesh elements have been obtained for a component of interest, the phononic stability of one or more identified mesh elements may be determined. Specifically, one or more mesh elements may be identified either prior to and/or after the finite element analysis has been conducted at 412. Depending on the particular embodiment, this identification may either be manually entered by a user using an appropriate input device and/or mesh elements that have been indicated as corresponding to a material that is sensitive to the application of strain may automatically be identified for evaluation of the material properties versus the determined strain states applied to those elements. For example, a meshed model may indicate that one or more elements are formed from a strain sensitive material such as silicon. After determining the applied strain states for the various mesh elements, those mesh elements of the meshed model that correspond to silicon, or another appropriate strain sensitive material, may be automatically identified for determining a stability, or other appropriate parameter, of the material.

Once the one or more mesh elements for determining a stability state have been identified, the strain states for the identified mesh elements may be provided as inputs to a trained phononic energy and/or frequency dispersion model at 414. One or more corresponding output stability states for the individual mesh elements may be output from the trained phononic energy and/or frequency dispersion bandgap model at 416. The output stability states for the various mesh elements may then be stored in an appropriate non-transitory computer readable medium for subsequent use and/or the output bandgaps may be used to update a stability parameter of the individual associated mesh elements at 418. In some embodiments, an indication of the stability of the one or more mesh elements may be output to a user in any appropriate fashion. For example, the determined stability of the mesh elements may be presented to a user in textual and/or graphical form. Alternatively, a graphical representation of the component model may be overlaid with an appropriate indication of the stability using indications such as color to indicate the corresponding stability within a particular portion of the modeled component. Of course, it should be understood that the current disclosure is not limited to how the information is output to a user, and in some instances, the information may not be output to a user. It should, of course, be understood that information regarding the stability of a component may be associated with information regarding the electronic properties (e.g., bandgap) of a component. For example, an additional (not pictured) step of the method may comprise providing the strain states for identified mesh elements as inputs to a trained statistical bandgap model. This may allow a user to obtain a bandgap associated with the identified mesh elements from the trained statistical bandgap model. In some embodiments, the method further comprises storing and/or outputting to a user an indication of bandgaps of the plurality of mesh elements.

While a single component has been discussed in relation to the above embodiment, embodiments in which a finite element analysis is applied to an overall assembly including the described component are also contemplated. For example, the component model within the overall assembly model may be identified as being made with a material that is sensitive to strain as described above. Thus, when the finite element analysis determines the strains for the various sub-parts within the assembly, including the component made from the strain sensitive material, a trained statistical model may be used to determine one or more properties of the component. Thus, the current disclosure may be applied to either individual components made from various materials of interest and/or to entire assemblies including multiple components as the disclosure is not limited in this fashion.

Again, while the above method has been described relative to the use and application of a trained statistical bandgap model, a Finite Element Analysis module may be used in combination with a trained statistical dispersion energy module as well. For example, the outputs provided by an electron energy dispersion model may include, but are not limited to, the dispersion energies of the material, a band structure of the material, transitions to between different states, and/or any other appropriate material property.

The above disclosed method provides for the simple, quick, and accurate determination of the both the strain state and corresponding changes in material properties and phase stability for a component which in combination provide a powerful tool for design purposes of elastic strain engineered components.

FIG. 9 illustrates a method of use 500 of a trained statistical model of a material. In the depicted embodiment, a strain state of a material may be obtained at 502. Again, the particular strain coordinates associated with the strain state may be obtained in any appropriate fashion, including input from a user, a strain state determined using a finite element analysis module, experimental data, an output of a trained statistical bandgap model (e.g., which has identified the strain state as associated with a desired bandgap) and/or any other appropriate source of the desired information. Regardless of how the strain state is obtained, it may be provided to a first statistical model (e.g., a trained statistical phononic energy and/or frequency dispersion model) in order to estimate one or more phonon frequencies associated with the strain state at 504. The first statistical model may be used to determine whether the strain state is a stable state at 505 using any of the above noted methods. If the strain state is stable at step 505, the strain state may optionally be provided to a second trained statistical model (e.g., an trained statistical bandgap model) at 506. At 507, the second trained statistical model may output the bandgap and/or the energy dispersion of the strained material. The output may be stored in a non-transitory computer readable medium for future use and/or the energy dispersion of material may be output to a user at 508. For example, the energy dispersion of the strained material may be plotted on a graph for visualization purposes of the band structure of the strained material by a user. Alternatively, in some embodiments, the output energy dispersion of the material may be used to identify one or more material properties of the material in the strained state at 510 as described further below. The one or more identified material properties may be stored on a non-transitory computer readable medium for future use and/or they may be output to a user at 512 using either textual and/or graphical presentations.

The above noted energy dispersion, i.e., band structure, of a material subjected to an elastic strain may be useful in multiple contexts. For example, simply plotting the energy dispersion of the strained material and outputting it to a user for visualization purposes may permit the user to study and explore the electronic behavior and properties of the elastically strained material using first-principal understandings of this material provided by such a visualization. Currently, such a process is unable to be done without extreme effort for every strain state of interest. In contrast, the disclosed method may be accomplished easily and quickly for a number of different possible strain states using the trained statistical models disclosed herein. Additionally, by generating the energy dispersion data associated with a material in an elastically strained state, it is possible to identify certain material property transitions and characteristics quickly and easily. For example, by examining where the conduction band minimum and the valence band maximum are located within the energy dispersion data, it is possible to identify the bandgap of the material, transitions from direct to indirect bandgaps, as well as providing information on various band properties such as the electron band, phonon band, magnon band, and other appropriate characteristics of the strained material through a straight forward analysis of the energy dispersion of the material. Again, this type of analysis and information may be either stored and/or output to a user. Alternatively, this type of analysis information may be combined with finite element analysis, and/or other appropriate types of analyses, to provide enhanced functionality relative to the information available to a user regarding the design and operation of a system.

In some embodiments, a trained statistical model (e.g., a phononic energy and/or frequency dispersion model) may be used to determine a phonon stability envelope. FIG. 10 schematically represents an exemplary stability (e.g., thermodynamic stability and/or mechanical stability) of a material over a cross-section 150 of a strain space, according to certain embodiments. Strain space 150 comprises axes ε_(i) and ε_(j), each of which corresponds to a component of the strain tensor (e.g., each of ε_(i) and ε_(j) corresponds to one of ε₁₁, ε₂₂, ε₃₃, ε₁₂, ε₁₃, or ε₂₃ and ε₁≠ε_(j)), in some embodiments. Strain space 150 further comprises a phononically stable region 152 corresponding to strains within the envelope, which includes an undeformed state 156 (wherein the strain tensor is zero), and a phononically unstable region 154 corresponding to strains outside the envelope, in some embodiments. According to some embodiments, multiaxial strains that falls within phononically stable region 152 is associated with thermodynamically stable strain states, whereas multiaxial strains that falls within phononically unstable region 154 is associated with thermodynamic instability and/or phase change of a material. According to some embodiments, a phononic stability envelope 158 associated with the transition between phononically stable region 152 and phononically unstable region 154 can be determined using a trained statistical model as detailed above (e.g., a phononic energy and/or frequency dispersion model). Thus, in some embodiments, any strain state within the phononic stability envelope may be thermodynamically stable. In some embodiments, the material is deformed such that the entire material remains within a phononically stable region of the strain space. This may be achieved, for example, by modeling strain at a plurality of points within a deformed configuration of the material and subsequently determining the phononic stability of the strain associated with the plurality of points. Typically, increasing strain tends to increase a probability of phononic instability within a material. For example, in FIG. 10, a probability of a strain falling within phononically stable region 152 decreases with increasing distance from undeformed state 156. However, phononic stability envelope 158 varies in distance from undeformed state 156, depending on the values of strain coordinates ε_(i) and ε_(j). Of course, it should be understood that FIG. 10 presents a cross-section of a phononic stability envelope. In some embodiments, a phononic stability envelope may be determined for a 3D, a 4D, a 5D, and/or a 6D strain-space. In some embodiments, a phonon stability envelope may be determined for a higher-dimensional strain-space (e.g., a 6D strain-space) and a cross-section of the phonon stability envelope may be determined for a lower-dimensional strain-space (e.g., a 2D or 3D cross-section of a 6D strain space may be determined). Exemplary representations of phonon stability envelopes in are provided in the examples below. It should be understood that the stability envelope may either be an approximated or fitted surface and/or may correspond to a plurality of spaced apart points in the multidimensional strain space as the disclosure is not limited in regards to how the envelope is implemented. Additionally, there are instances in which a stability envelope may be previously determined using a trained statistical model using the methods described herein and may be used in combination with a finite element modeling module and/or bandgap determination module as described above to ensure the strain states either used in, or output from, the finite element model and/or bandgap model are associated with strains within the strain states contained in the interior of the stability envelope. Bandgaps and/or strain states from a finite element model may then be output, stored, and/or used in any of the ways detailed further above in regards to the other embodiments.

The above-described embodiments of the technology described herein can be implemented in any of numerous ways. For example, the embodiments may be implemented using hardware, software or a combination thereof. When implemented in software, the software code can be executed on any suitable processor or collection of processors, whether provided in a single computing device or distributed among multiple computing devices. Such processors may be implemented as integrated circuits, with one or more processors in an integrated circuit component, including commercially available integrated circuit components known in the art by names such as CPU chips, GPU chips, microprocessor, microcontroller, or co-processor. Alternatively, a processor may be implemented in custom circuitry, such as an ASIC, or semicustom circuitry resulting from configuring a programmable logic device. As yet a further alternative, a processor may be a portion of a larger circuit or semiconductor device, whether commercially available, semi-custom or custom. As a specific example, some commercially available microprocessors have multiple cores such that one or a subset of those cores may constitute a processor. Though, a processor may be implemented using circuitry in any suitable format.

Further, it should be appreciated that a computing device may be embodied in any of a number of forms, such as a rack-mounted computer, a desktop computer, a laptop computer, or a tablet computer. Additionally, a computing device may be embedded in a device not generally regarded as a computing device but with suitable processing capabilities, including a Personal Digital Assistant (PDA), a smart phone, tablet, or any other suitable portable or fixed electronic device.

Also, a computing device may have one or more input and output devices. These devices can be used, among other things, to present a user interface. Examples of output devices that can be used to provide a user interface include display screens for visual presentation of output and speakers or other sound generating devices for audible presentation of output. Examples of input devices that can be used for a user interface include keyboards, individual buttons, and pointing devices, such as mice, touch pads, and digitizing tablets. As another example, a computing device may receive input information through speech recognition or in other audible format.

Such computing devices may be interconnected by one or more networks in any suitable form, including as a local area network or a wide area network, such as an enterprise network or the Internet. Such networks may be based on any suitable technology and may operate according to any suitable protocol and may include wireless networks, wired networks or fiber optic networks.

Also, the various methods or processes outlined herein may be coded as software that is executable on one or more processors that employ any one of a variety of operating systems or platforms. Additionally, such software may be written using any of a number of suitable programming languages and/or programming or scripting tools, and also may be compiled as executable machine language code or intermediate code that is executed on a framework or virtual machine.

In this respect, the embodiments described herein may be embodied as a computer readable storage medium (or multiple computer readable media) (e.g., a computer memory, one or more floppy discs, compact discs (CD), optical discs, digital video disks (DVD), magnetic tapes, flash memories, RAM, ROM, EEPROM, circuit configurations in Field Programmable Gate Arrays or other semiconductor devices, or other tangible computer storage medium) encoded with one or more programs that, when executed on one or more computers or other processors, perform methods that implement the various embodiments discussed above. As is apparent from the foregoing examples, a computer readable storage medium may retain information for a sufficient time to provide computer-executable instructions in a non-transitory form. Such a computer readable storage medium or media can be transportable, such that the program or programs stored thereon can be loaded onto one or more different computing devices or other processors to implement various aspects of the present disclosure as discussed above. As used herein, the term “computer-readable storage medium” encompasses only a non-transitory computer-readable medium that can be considered to be a manufacture (i.e., article of manufacture) or a machine. Alternatively or additionally, the disclosure may be embodied as a computer readable medium other than a computer-readable storage medium, such as a propagating signal.

The terms “program” or “software” are used herein in a generic sense to refer to any type of computer code or set of computer-executable instructions that can be employed to program a computing device or other processor to implement various aspects of the present disclosure as discussed above. Additionally, it should be appreciated that according to one aspect of this embodiment, one or more computer programs that when executed perform methods of the present disclosure need not reside on a single computing device or processor, but may be distributed in a modular fashion amongst a number of different computers or processors to implement various aspects of the present disclosure.

Computer-executable instructions may be in many forms, such as program modules, executed by one or more computers or other devices. Generally, program modules include routines, programs, objects, components, data structures, etc. that perform particular tasks or implement particular abstract data types. Typically the functionality of the program modules may be combined or distributed as desired in various embodiments.

The embodiments described herein may be embodied as a method, of which an example has been provided. The acts performed as part of the method may be ordered in any suitable way. Accordingly, embodiments may be constructed in which acts are performed in an order different than illustrated, which may include performing some acts simultaneously, even though shown as sequential acts in illustrative embodiments.

Further, some actions are described as taken by a “user.” It should be appreciated that a “user” need not be a single individual, and that in some embodiments, actions attributable to a “user” may be performed by a team of individuals and/or an individual in combination with computer-assisted tools or other mechanisms.

With reference to FIG. 11, an exemplary system for implementing aspects of the invention includes a general purpose computing device in the form of a computer 610. Components of computer 610 may include, but are not limited to, a processing unit 620, a system memory 630, and a system bus 621 that couples various system components including the system memory to the processing unit 620. The system bus 621 may be any of several types of bus structures including a memory bus or memory controller, a peripheral bus, and a local bus using any of a variety of bus architectures. By way of example, and not limitation, such architectures include Industry Standard Architecture (ISA) bus, Micro Channel Architecture (MCA) bus, Enhanced ISA (EISA) bus, Video Electronics Standards Association (VESA) local bus, and Peripheral Component Interconnect (PCI) bus also known as Mezzanine bus.

Computer 610 typically includes a variety of computer readable media. Computer readable media can be any available media that can be accessed by computer 610 and includes both volatile and nonvolatile media, removable and non-removable media. By way of example, and not limitation, computer readable media may comprise computer storage media and communication media. Computer storage media includes both volatile and nonvolatile, removable and non-removable media implemented in any method or technology for storage of information such as computer readable instructions, data structures, program modules or other data. Computer storage media includes, but is not limited to, RAM, ROM, EEPROM, flash memory or other memory technology, CD-ROM, digital versatile disks (DVD) or other optical disk storage, magnetic cassettes, magnetic tape, magnetic disk storage or other magnetic storage devices, or any other medium which can be used to store the desired information and which can accessed by computer 610. Communication media typically embodies computer readable instructions, data structures, program modules or other data in a modulated data signal such as a carrier wave or other transport mechanism and includes any information delivery media. The term “modulated data signal” means a signal that has one or more of its characteristics set or changed in such a manner as to encode information in the signal. By way of example, and not limitation, communication media includes wired media such as a wired network or direct-wired connection, and wireless media such as acoustic, RF, infrared and other wireless media. Combinations of the any of the above should also be included within the scope of computer readable media.

The system memory 630 includes computer storage media in the form of volatile and/or nonvolatile memory such as read only memory (ROM) 631 and random access memory (RAM) 632. A basic input/output system 633 (BIOS), containing the basic routines that help to transfer information between elements within computer 610, such as during start-up, is typically stored in ROM 631. RAM 632 typically contains data and/or program modules that are immediately accessible to and/or presently being operated on by processing unit 620. By way of example, and not limitation, FIG. 11 illustrates operating system 634, application programs 635, other program modules 636, and program data 637.

The computer 610 may also include other removable/non-removable, volatile/nonvolatile computer storage media. By way of example only, FIG. 11 illustrates a hard disk drive 641 that reads from or writes to non-removable, nonvolatile magnetic media, a magnetic disk drive 651 that reads from or writes to a removable, nonvolatile magnetic disk 652, and an optical disk drive 655 that reads from or writes to a removable, nonvolatile optical disk 656 such as a CD ROM or other optical media. Other removable/non-removable, volatile/nonvolatile computer storage media that can be used in the exemplary operating environment include, but are not limited to, magnetic tape cassettes, flash memory cards, digital versatile disks, digital video tape, solid state RAM, solid state ROM, and the like. The hard disk drive 641 is typically connected to the system bus 621 through an non-removable memory interface such as interface 640, and magnetic disk drive 651 and optical disk drive 655 are typically connected to the system bus 621 by a removable memory interface, such as interface 650.

The drives and their associated computer storage media discussed above and illustrated in FIG. 11, provide storage of computer readable instructions, data structures, program modules and other data for the computer 610. In FIG. 11, for example, hard disk drive 641 is illustrated as storing operating system 644, application programs 645, other program modules 646, and program data 647. Note that these components can either be the same as or different from operating system 634, application programs 635, other program modules 636, and program data 637. Operating system 644, application programs 645, other program modules 646, and program data 647 are given different numbers here to illustrate that, at a minimum, they are different copies. A user may enter commands and information into the computer 610 through input devices such as a keyboard 662 and pointing device 661, commonly referred to as a mouse, trackball or touch pad. Other input devices (not shown) may include a microphone, joystick, game pad, satellite dish, scanner, or the like. These and other input devices are often connected to the processing unit 620 through a user input interface 660 that is coupled to the system bus, but may be connected by other interface and bus structures, such as a parallel port, game port or a universal serial bus (USB). A monitor 691 or other type of display device is also connected to the system bus 621 via an interface, such as a video interface 690. In addition to the monitor, computers may also include other peripheral output devices such as speakers 697 and printer 696, which may be connected through a output peripheral interface 695.

The computer 610 may operate in a networked environment using logical connections to one or more remote computers, such as a remote computer 680. The remote computer 680 may be a personal computer, a server, a router, a network PC, a peer device or other common network node, and typically includes many or all of the elements described above relative to the computer 610, although only a memory storage device 681 has been illustrated in FIG. 11. The logical connections depicted in FIG. 11 include a local area network (LAN) 671 and a wide area network (WAN) 673, but may also include other networks. Such networking environments are commonplace in offices, enterprise-wide computer networks, intranets and the Internet.

When used in a LAN networking environment, the computer 610 is connected to the LAN 671 through a network interface or adapter 670. When used in a WAN networking environment, the computer 610 typically includes a modem 672 or other means for establishing communications over the WAN 673, such as the Internet. The modem 672, which may be internal or external, may be connected to the system bus 621 via the user input interface 660, or other appropriate mechanism. In a networked environment, program modules depicted relative to the computer 610, or portions thereof, may be stored in the remote memory storage device. By way of example, and not limitation, FIG. 11 illustrates remote application programs 685 as residing on memory device 681. It will be appreciated that the network connections shown are exemplary and other means of establishing a communications link between the computers may be used.

EXAMPLE 1

Phonon stability is the minimal requirement for lattice stability and loading reversibility. Whether mechanically strained or not, the absence of non-real and/or negative phonon frequency for the wavevector in the entire Brillouin zone is the hallmark of a locally stable crystal lattice. If a strained perfect crystal lattice has a stable phonon band structure, then at T=0 K and in the absence of defects such as free surfaces, interfaces and dislocations, this lattice is guaranteed to avoid spontaneous phase transition or defect nucleation. Consequently, phonon stability is the minimal requirement for lattice stability and loading reversibility. This example describes a method for computing phononic stability of a given stress state in undoped diamond, to determine the stability of a given stress state. The phononic band structures of diamond under tensorial strain were predicted ab initio, using a machine learning (ML) model of phononic band structures with high accuracy based on ab initio density functional theory (DFT).

In this example, the ML algorithm was applied to fit the phonon dispersion of a given material, using training data computed at a variety of the 6D strain values. Training data were produced using phonon calculations that were mainly conducted using the VASP-Phonopy package. 2×2×2 supercells of 16 carbon atoms were created, and phonon calculations were conducted with a 3×3×3 k-point mesh. Whenever accurate phonon stability check was needed for diamond primitive cell, DFPT [160] as implemented in Quantum ESPRESSO [161] was adopted, with a dense 11×11×11 k-grid and 6×6×6 q-grid.

Since phononic stability depends on the absence of non-real and/or negative frequency values, the trained ML model could be used to identify a stability envelope within strain space. FIG. 12 presents the phonon density of states (DOS), computed at 3 different strains, comparing results of the ML model and first principles calculations (labeled “ground true”). In the top and bottom examples, all frequencies are real and non-negative, resulting in a zero density of states at negative frequencies. When non-real or negative frequencies are observed, the density of states function becomes nonzero at negative frequencies, as indicated by feature 900 of FIG. 12. Thus, the strains associated with the top and bottom phonon density of states functions presented in FIG. 12 are considered to be phononically stable, whereas strains producing features such as feature 900 are considered phononically unstable. Generally, while the ML model does not perfectly match the results of first principles calculations, the DOS functions produced by first principles and the ML model broadly agree on the major structural features of the DOS functions presented in FIG. 12, indicating the viability of this approach for phononic stability calculations.

Using the ML model, a phonon stability envelope was calculated for the 6 dimensional strain space of diamond. By combining this envelope with a trained statistical bandgap model, the graphs represented in FIGS. 13A-13B were produced. FIG. 13A presents a 3D cross-section of strain space, indicating with voxels the positions of phononically stable strains within a phononically stable region 1152 determined using the methods detailed above. Each voxel is shaded according to the state of the diamond at that strain. As indicated in FIG. 13A, dark shaded voxels correspond to metallized strains. Projected below the phononically stable region 1152 is a smoothed rendering of a cross-section of phononically stable region 1152 collected at ε₃₃=−0.056 indicated by the shaded rectangle. Regions of this cross-section corresponding to different states of the electronic bandgap (e.g., direct metal, direct bandgap, indirect bandgap, and indirect metal) are indicated using labeled curves within the projection of the cross-section.

FIG. 13B presents the projected cross-section of FIG. 13A, in greater detail. Superimposed on this cross-section are a collection of strain points at which the electronic bandgap was calculated, allowing determination of the boundaries between regions having stable strain but different electronic bandgap states. Here, the phononic stability envelope is indicated by the dashed lines.

FIG. 14 presents a set of 2D cross-sections of the 6D strain space. This plot is made up of snapshots that characterize the shape of the stability envelope in 2D subspaces in the 30 off-diagonal graphics and of histograms in 6 diagonal graphics. Symmetries due to crystal deformation are obeyed when rendering this plot through ML models.

As indicated by this example, the phononic stability of a given strain may be determined using calculation methods described herein. In conjunction with estimates of strain on a material, this approach may therefore be used to identify the thermodynamic stability of a material under a given multiaxial strain. It should, of course, be understood that although the approaches taken in this example were applied to undoped diamond, an identical approach can taken using any material, and the disclosure is not so limited.

EXAMPLE 2

This example demonstrates that in some cases, phononic instability may occur before a 0 eV electronic bandgap occurs in undoped diamond, exclusively through the imposition of multiaxial elastic strains. This example demonstrates the value of modeling stability of a component. This discovery implies that reversible metallization/de-metallization is feasible through judicious design of mechanical loading conditions and geometry in nanoscale diamond. However, the example also demonstrates that further bending of the nanoneedle can however induce phonon instabilities that lead to irreversible sp³→sp² (diamond to graphite) phase transition or fracture.

Diamond nanoneedles may exhibit ultra-large elastic bending before fracture. In this example, deflection of diamond nanoneedles by an actuator (a diamond indenter), was simulated to demonstrate the role that multiaxial strain may play in metallizing the diamond nanoneedles. The presented schematically in FIG. 15, which show a diamond nanoneedle 800 deformed by contact with diamond indenter 802, which acts as an actuator, was simulated.

Such deformation, resulting in local compressive strains larger than −10% and tensile strains in excess of 9%, is reversible upon release of the load. Here simulations were applied to determine electronic bandgap modulation in bent diamond nanoneedles at maximum local strain levels that are known to be experimentally feasible. Initially, strain of the deformed nanoneedle was determined by finite element method (FEM) calculations used to simulate the sideward bending moment of the diamond needle upon contact with the indenter tip and to account for nonlinear elasticity, the orientation of the cubic lattice with respect to the needle axis, the bending direction, and possible friction between the indenter tip and the needle. Various degrees of deformation of the diamond nanoneedle were simulated, producing different spatial distributions of strain.

FIG. 16A shows FEM predictions of local compressive and tensile strains of a deformed geometry of <110>diamond nanoneedle, with the maximum compressive and tensile strains of −10.8% and 9.6% respectively.

The electronic band structures of diamond under tensorial strain were predicted ab initio, using a machine learning (ML) model of electronic band structures with high accuracy based on ab initio density functional theory (DFT) followed by GW calculations. (GW calculations are made using the GW approximation, where the system to (or from) which the electron is added (or removed) is described as a polarizable, screening medium. This is expressed by the name of the approximation: ‘GW’ stands for the one-particle Green's function G and for W, the dynamically screened Coulomb interaction.) Because GW calculations are computationally expensive, in this example a stress-strain constitutive law for modeling large elastic deformation of diamond in any arbitrary sample geometry, along with fast proxy models for the electronic and phonon band structures was used to produce a machine learning (ML) model of band structures based on GW approximation training data, so as to perform coupled ab initio and finite element calculations with constitutive laws based on neural nets.

Thus, in addition to the FEM analysis of the diamond nanoneedle, FIG. 16A presents the corresponding predictions of the spatial distribution of the electronic bandgap of the diamond. The onset of metallization appears in the severely strained compressive side of the nanoneedle at a local strain of −10.8%, as shown in FIG. 16A.

Additionally, in this example, phononic stability analysis was used to determine the maximum extent to which stable deflection could be supported by the diamond nanoneedle, using a phononic stability model described in Example 1, above.

FIG. 16B presents the diamond nanoneedle of FIG. 16B under the largest deflection observed to be phononically stable. The <110>nanoneedle was able to withstand up to 12.1% local tensile strain before incurring phonon instability on the tensile side, at an electronic bandgap of 0.62 eV, as shown in FIG. 16B. The compressive side was more tolerant to deformation. The maximum attainable compressive strain could be on the order of −20% along a low-index orientation [133], suggesting that there was room for additional elastic deformation after achieving “safe” metallization in compression-dominated regions.

FIGS. 17A-17B present the lowest electronic bandgap of the diamond nanoneedles expected as a function of compressive (FIG. 17A) and tensile (FIG. 17B) stress. Under compressive stress, the diamond nanoneedles transitioned from a non-conducting state (under zero strain) to a “safe” metallic state, as indicated by the reduction of the electronic bandgap to zero in FIG. 17A. By contrast, in FIG. 17B, while tensile strain reduced the electronic bandgap of the diamond nanoneedle, the electronic bandgap did not reach zero before a phononic instability was encountered (indicated by the labeled point) that would result in graphitization of the diamond under tension. Nonetheless, the diamond nanoneedle still experienced a transition from a non-conducting state to a semi-conducting state as simulated tensile strain increased.

Different simulations of deflection of the diamond nanoneedle, with different coefficients of friction between the nanoneedle and the actuator, indicated that the tendency to metallize as overall strain increased was independent of friction between the actuator and the diamond nanoneedle. Note that due to the zero-point motion effect and the Varshni effect, for physical experiments performed at room temperature, the electronic bandgap of diamond is expected to be even smaller than estimated here by 0.4-0.6 eV. This understanding leads to the inference that safe metallization in diamond can occur at elastic strain levels somewhat smaller than indicated by these calculations, making metallization more easily achievable (and/or achievable at lower strains) than the results in FIGS. 17A-17B suggest.

This example further demonstrates the spatial evolution of the “safe” direct bandgap regions (i.e., regions of the material associated with phononically stable strains and direct bandgap transitions) in the nanoneedles can be found in FIG. 18. In particular, FIG. 18 presents the volume of the direct-bandgap region of a <111>oriented needle, indicated by star-shaped data points, alongside the minimum bandgap observed within the needle, indicated by circular data points. As expected, as strain increased, the volume of the direct bandgap region increased and the minimum electronic bandgap energy decreased until it reached a value near zero. The graphitized region indicated in the figure reflects the onset of phononic instability. It should, of course, be understood that although the approaches taken in this example were applied to undoped diamond, an identical approach can taken using any material, and the disclosure is not so limited.

While the present teachings have been described in conjunction with various embodiments and examples, it is not intended that the present teachings be limited to such embodiments or examples. On the contrary, the present teachings encompass various alternatives, modifications, and equivalents, as will be appreciated by those of skill in the art. Accordingly, the foregoing description and drawings are by way of example only.

While several embodiments of the present invention have been described and illustrated herein, those of ordinary skill in the art will readily envision a variety of other means and/or structures for performing the functions and/or obtaining the results and/or one or more of the advantages described herein, and each of such variations and/or modifications is deemed to be within the scope of the present invention. More generally, those skilled in the art will readily appreciate that all parameters, dimensions, materials, and configurations described herein are meant to be exemplary and that the actual parameters, dimensions, materials, and/or configurations will depend upon the specific application or applications for which the teachings of the present invention is/are used. Those skilled in the art will recognize, or be able to ascertain using no more than routine experimentation, many equivalents to the specific embodiments of the invention described herein. It is, therefore, to be understood that the foregoing embodiments are presented by way of example only and that, within the scope of the appended claims and equivalents thereto, the invention may be practiced otherwise than as specifically described and claimed. The present invention is directed to each individual feature, system, article, material, kit, and/or method described herein. In addition, any combination of two or more such features, systems, articles, materials, kits, and/or methods, if such features, systems, articles, materials, kits, and/or methods are not mutually inconsistent, is included within the scope of the present invention. 

What is claimed is:
 1. A method comprising, by at least one processor: obtaining a range of strain coordinates; obtaining training data, wherein the training data includes phononic energy and/or frequency dispersion training data for a material within the range of strain coordinates; generating a first statistical model using the training data and the range of strain coordinates, wherein the first statistical model is associated with the material and configured to output phononic energy and/or frequency dispersion values using at least one strain coordinate; and storing the first statistical model.
 2. The method of claim 1, wherein the strain coordinates have at least three degrees of freedom.
 3. The method of claim 1, wherein the training data comprise a plurality of training data points, each training data point is associated with an energy and/or frequency value in a range of reciprocal space and a corresponding phonon density of states (DOS) value.
 4. At least one non-transitory computer-readable storage medium storing processor executable instructions that, when executed by at least one processor, cause the at least one processor to perform the method of claim
 1. 5. A system comprising: a non-transitory computer-readable storage medium storing programming instructions; and at least a processor configured to execute the programming instructions to perform operations comprising the method of claim
 1. 6. A method, comprising, by at least one processor: obtaining a strain state of a component; using a first statistical model to estimate one or more phonon frequencies associated with the strain-state; and determining whether the strain state is a stable state using the estimated one or more phonon frequencies.
 7. The method of claim 6, further comprising determining at least one phonon density of states (DOS) value associated with one of the estimated one or more phonon frequencies.
 8. The method of claim 6, wherein determining whether the strain state is a stable state comprises: determining whether any of the estimated one or more phonon frequencies has a non-real or negative value; or determining that the strain state is a stable state if none of the estimated one or more phonon frequencies has a non-real or negative value.
 9. The method of claim 8, further comprising determining a phonon stability envelope in which any strain state is a stable state.
 10. The method of claim 6, wherein obtaining the strain state of the component comprises: determining the strain state of the component, at least in part by determining a strain state of a mesh element of a plurality of mesh elements in a mesh model of the component.
 11. The method of claim 6, further comprising: obtaining a bandgap corresponding to the strain state of the component using a second statistical model.
 12. The method of claim 11, wherein the second statistical model is at least one selected from the group of a bandgap model and dispersion energy model of the material.
 13. The method of claim 10, further comprising outputting an indication of bandgaps and phononic stability of the plurality of mesh elements.
 14. At least one non-transitory computer-readable storage medium storing processor executable instructions that, when executed by at least one processor, cause the at least one processor to perform the method of claim
 6. 15. A system comprising: a non-transitory computer-readable storage medium storing programming instructions; and at least a processor configured to execute the programming instructions to perform operations comprising the method of claim
 6. 16. A method comprising: providing a desired bandgap to a trained statistical bandgap model of a material to obtain one or more strain states corresponding to the desired bandgap; using a first statistical model to estimate one or more phonon frequencies associated with at least one of the one or more strain states; and determining whether at least one of the one or more strain states is a stable state using the estimated one or more phonon frequencies.
 17. The method of claim 16, further comprising identifying a stable strain state with a lowest strain energy density associated with the desired bandgap.
 18. The method of claim 16, wherein determining whether the strain state is a stable state comprises: determining whether any of the estimated one or more phonon frequencies has a non-real or negative value; and determining that the strain state is a stable state if none of the estimated one or more phonon frequencies has a non-real or negative value.
 19. At least one non-transitory computer-readable storage medium storing processor executable instructions that, when executed by at least one processor, cause the at least one processor to perform the method of claim
 16. 20. A system comprising: a non-transitory computer-readable storage medium storing programming instructions; and at least a processor configured to execute the programming instructions to perform operations comprising the method of claim
 16. 